1. Field of the Invention
The present invention relates in general to a method for optimally determining sensor positions for an acoustic pyrometry, and more particularly to such a method which is capable of obtaining a minimum number of sensors satisfying a given error limit on the basis of an effective independence (EfI) method that is occasionally used for an acoustic holography or Indirect input measurement and determining positions of the obtained number of sensors suitable for error minimization.
2. Description of the Prior Art
As well known in the art, an acoustic pyrometry is a method useful to real-time measurement of temperature distributions on a two-dimensional sectional area. This pyrometry can be particularly useful in measuring a spatial temperature distribution and its time-based variations in a high-temperature place where a boiler is located or a harmful environment where a chemical reactor is located. For example, information regarding temperature distributions in a boiler used in a power plant is required in determining whether the boiler is abnormally operated and normally operating the boiler.
For measurement of temperature distributions in a boiler using the acoustic pyrometry, a plurality of acoustic emitters (also used as sensors) are mounted on the sidewall of the boiler and each of them senses sounds emitted from the others. Then, a sound propagation time based on the combination of the sensors is measured and subjected to an appropriate mathematical calculation process, thereby measuring a two-dimensional temperature distribution within the boiler, not a temperature at one point. At this time, an error of the measured temperature distribution is greatly influenced by the number and positions of the used sensors.
Now, a detailed description will be given of the basic principle of the above-stated acoustic pyrometry.
The acoustic pyrometry is a method for measuring a temperature in space using the fact that a propagation velocity of a sound is a function of the temperature. Namely, assuming that given conditions are a normal state, isentropic process, ideal gas, etc., it is well known from a continuity equation, momentum equation and state equation that the relation of the following equation 1 exists between the propagation velocity of the sound and the temperature.
xe2x80x83C={square root over (xcex3RT)}=K{square root over (T)}xe2x80x83xe2x80x83[Equation 1]
where, C=acoustic velocity, T=temperature of medium [K],   K  =                    γ        ⁢                  xe2x80x83                ⁢        R              =                                                      c              p                        ⁢                          R              0                                                          c              v                        ⁢            M                          ,            
xe2x80x83xcex3=specific heat ratio, R0=gas constant and M=molecular mass.
Using the expression of the acoustic velocity in the above equation 1, it can be seen that the following equation 2 is established between a travel time t and travel distance dl of a sound pulse propagated within a sound field.                     t        =                                            ⅆ              l                                      C              ⁡                              (                                  x                  ,                  y                                )                                              =                                    ⅆ              l                                      K              ⁢                                                T                  ⁡                                      (                                          x                      ,                      y                                        )                                                                                                          [                  Equation          ⁢                      xe2x80x83                    ⁢          2                ]            
FIG. 1 is a graph illustrating the relation between sensor positions and a sound propagation path. Assuming that sensors are present respectively at two points A(x0, y0) and B(x1, y1) in a rectangular sectional area as shown in FIG. 1, calculation can be made with respect to time required for an acoustic wave emitted from the sensor at the point A to arrive at the sensor at the point B. Although the actual travel path of the acoustic wave is slightly curved from a straight line between the two points A and B because it is refracted due to a temperature gradient, it will be assumed to be a straight line. Hence, an arbitrary point on the path between the two points A and B can be expressed as in the following equation 3.
x(s)=x0+(x1xe2x88x92x0)s
y(s)=y0+(y1xe2x88x92y0)sxe2x80x83xe2x80x83[Equation 3]
where,                               s          =                      l            L                          ,                            [                  Equation          ⁢                      xe2x80x83                    ⁢          3                ]            
xe2x80x83L={square root over ((x0xe2x88x92x1)2+(y0xe2x88x92y1)2)} (0xe2x89xa6sxe2x89xa61).
In the above equation 3, L is the distance between the two points A and B and l is the distance between the arbitrary point on the path and the point A. Dividing the x-coordinate system and y-coordinate system in the equation 3 respectively by lengths X and Y, dimensionless coordinate systems u and v can be obtained as in the below equation 4.                               u          =                      x            X                          ,                  v          =                      y            Y                                              [                  Equation          ⁢                      xe2x80x83                    ⁢          4                ]            
where, u(s)=u0+(u1xe2x88x92u0)s, v(s)=v0+(v1xe2x88x92v0)s (0xe2x89xa6sxe2x89xa61).
Because the temperature is a function of the position (x,y), or (u,v), the acoustic velocity is defined as a function of the position, too.
Accordingly, substituting (x, y) in the above equation 2 with (u, v) and integrating both sides on the basis of ds (or dl), the travel time of the sound can be expressed as in the following equation 5.                               t          =                                    L              ⁢                                                ∫                  0                  1                                ⁢                                                      ⅆ                    s                                                        C                    ⁡                                          (                                              u                        ,                        v                                            )                                                                                            =                          L              ⁢                                                ∫                  0                  1                                ⁢                                                      F                    ⁡                                          (                                              u                        ,                        v                                            )                                                        ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    s                                                                                      ⁢                  xe2x80x83                                    [                  Equation          ⁢                      xe2x80x83                    ⁢          5                ]            
where,       xe2x80x83    ⁢            F      ⁢              (                  u          ,          v                )              =                  1                  C          ⁡                      (                          u              ,              v                        )                              .      
Expressing F(u, v) in the above equation 5 as a Fourier series, the result is:                               F          ⁡                      (                          u              ,              v                        )                          =                              1                          K              ⁢                                                T                  ⁡                                      (                                          u                      ,                      v                                        )                                                                                =                                    ∑              m                        ⁢                          xe2x80x83                        ⁢                                          ∑                n                            ⁢                                                A                  mn                                ⁢                                  G                  ⁡                                      (                    mu                    )                                                  ⁢                                  H                  ⁡                                      (                    nv                    )                                                                                                          [                  Equation          ⁢                      xe2x80x83                    ⁢          6                ]            
In the above equation 6, G and H are admissible functions, which can be defined respectively as G=cos(mu) and H=cos(nv) in a rectangular sectional area. Substituting the equation 5 with the equation 6, the result is:                     t        =                              ∑            m                    ⁢                      xe2x80x83                    ⁢                                    ∑              n                        ⁢                                          A                mn                            ⁢                              f                mn                                                                        [                  Equation          ⁢                      xe2x80x83                    ⁢          7                ]            
where, fmn=L ∫ G(mu)H(nv)ds, and m and n are dummy indexes.
Assuming that the number of sensors is N and the number of paths of each of the sensors is p, respective travel times t can be experimentally obtained as p simultaneous equations (p equations 7), which can be expressed in matrix as in the below equation 8.
[f]pxc3x97q{A}qxc3x97l={t}pxc3x97lxe2x80x83xe2x80x83[Equation 8]
In the above equation 8, {A}={A00, A01, A10, A02, A11, A20, . . . }, and q is the number of coefficient terms taken for sufficient convergence of the Fourier coefficient A and must be smaller than or equal to p.
A Fourier coefficient vector {A} can be obtained from the measured time as in the following equation 9.
{[f]T}qxc3x97p[f]pxc3x97q{A}qxc3x97l={[f]T}qxc3x97p{t}pxc3x97l{A}qxc3x97l={{[f]T}qxc3x97p[f]pxc3x97q}qxc3x97qxe2x88x921{{[f]T}qxc3x97p{t}pxc3x97l}qxc3x97lxe2x80x83xe2x80x83[Equation 9]
For example, for temperature measurement on a rectangular plane, fmn is expressed as in the following equation 10.                                                                                                                                                             1                        )                                            ⁢                                              xe2x80x83                                            ⁢                      If                      ⁢                                              xe2x80x83                                            ⁢                      m                                        =                                          n                      =                      0                                                        ,                                      xe2x80x83                                    ⁢                                                            f                      m                                        =                                          L                      .                                              
                                            ⁢                      2                                                                      )                            ⁢                              xe2x80x83                            ⁢              If              ⁢                              xe2x80x83                            ⁢                              m                ⁡                                  (                                                            u                      1                                        -                                          u                      0                                                        )                                                      -                          n              ⁡                              (                                                      v                    1                                    -                                      v                    0                                                  )                                              =          0                ,                  
                ⁢                              f            mn                    =                                                                      L                  2                                ⁢                                  xe2x80x83                                ⁢                                                      {                                                                  cos                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    mu                              0                                                        -                                                          nv                              0                                                                                )                                                                    +                                                                                                    sin                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                mu                                  1                                                                +                                                                  nv                                  1                                                                                            )                                                                                -                                                      sin                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                mu                                  0                                                                +                                                                  nv                                  0                                                                                            )                                                                                                                                                            m                            ⁡                                                          (                                                                                                u                                  1                                                                -                                                                  u                                  0                                                                                            )                                                                                +                                                      n                            ⁡                                                          (                                                                                                v                                  1                                                                -                                                                  v                                  0                                                                                            )                                                                                                                                            }                                    .                                      "IndentingNewLine"                                    ⁢                  3                                ⁢                                  )                                ⁢                                  xe2x80x83                                ⁢                If                ⁢                                  xe2x80x83                                ⁢                                  m                  ⁡                                      (                                                                  u                        1                                            -                                              u                        0                                                              )                                                              +                              n                ⁡                                  (                                                            v                      1                                        -                                          v                      0                                                        )                                                      =            0                          ,                  
                ⁢                              f            mn                    =                                    L              2                        ⁢                          xe2x80x83                        ⁢                                          {                                                      cos                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        mu                          0                                                +                                                  nv                          0                                                                    )                                                        +                                                                                    sin                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    mu                              1                                                        -                                                          nv                              1                                                                                )                                                                    -                                              sin                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                                                    mu                              0                                                        -                                                          nv                              0                                                                                )                                                                                                                                    m                        ⁡                                                  (                                                                                    u                              1                                                        -                                                          u                              0                                                                                )                                                                    -                                              n                        ⁡                                                  (                                                                                    v                              1                                                        -                                                          v                              0                                                                                )                                                                                                                    }                            .                              
                            ⁢              4                        ⁢                          )                        ⁢                          xe2x80x83                        ⁢            In            ⁢                          xe2x80x83                        ⁢            other            ⁢                          xe2x80x83                        ⁢            cases                          ,                  
                ⁢                              f            mn                    =                                                    L                2                            ⁢                              xe2x80x83                            ⁢                              {                                                                            sin                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              mu                            1                                                    +                                                      nv                            1                                                                          )                                                              -                                          sin                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                                              mu                            0                                                    +                                                      nv                            0                                                                          )                                                                                                                        m                      ⁡                                              (                                                                              u                            1                                                    -                                                      u                            0                                                                          )                                                              +                                          n                      ⁡                                              (                                                                              v                            1                                                    -                                                      v                            0                                                                          )                                                                                            }                                      +                                                                                sin                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        mu                          1                                                -                                                  nv                          1                                                                    )                                                        -                                      sin                    ⁢                                          xe2x80x83                                        ⁢                                          (                                                                        mu                          0                                                -                                                  nv                          0                                                                    )                                                                                                            m                    ⁡                                          (                                                                        u                          1                                                -                                                  u                          0                                                                    )                                                        -                                      n                    ⁡                                          (                                                                        v                          1                                                -                                                  v                          0                                                                    )                                                                                  .                                                          [                  Equation          ⁢                      xe2x80x83                    ⁢          10                ]            
As a result, the temperature distribution T(u,v) in the above equation 6 can be expressed as in the below equation 11.                               T          ⁡                      (                          u              ,              v                        )                          =                  1                                                    {                                  K                  ⁢                                      ∑                                          ∑                                                                        A                          mn                                                ⁢                        cos                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                          mu                          )                                                ⁢                                                  xe2x80x83                                                ⁢                        cos                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                          nv                          )                                                                                                                    }                            2                        ⁢                          xe2x80x83                                                          [                  Equation          ⁢                      xe2x80x83                    ⁢          11                ]            
The two-dimensional temperature distribution can be measured in the above manner. At this time, an error of the measured temperature distribution is greatly influenced by the number and positions of sensors. That is, different temperatures are measured according to positions of sensors and the measured temperature values are different in accuracy according to the number of the sensors. In the worst case, a temperature distribution to be measured may not be found or the measured results of the temperature distribution may express preposterous values. In this regard, how many sensors will be used and how the sensors will be arranged become issues. However, approaches to such issues have not been proposed by conventional studies and patents.
Therefore, the present invention has been made in view of the above problems, and it is an object of the present invention to provide a method for optimally determining sensor positions for an acoustic pyrometry, which is capable of obtaining a minimum number of sensors satisfying a given error limit on the basis of an effective independence method that is occasionally used for an acoustic holography or indirect input measurement and determining positions of the obtained number of sensors suitable for error minimization, so that temperature distributions can accurately be measured.
In accordance with the present invention, the above and other objects can be accomplished by a provision of a method for optimally determining sensor positions for an acoustic pyrometry, comprising the first step of arbitrarily arranging a predetermined number of sensors on sides of a rectangular sectional area to be measured; the second step of calculating a transfer matrix with respect to positions of all of the sensors; the third step of obtaining effective independence values of respective paths of each of the sensors by performing singular value decomposition with respect to the calculated transfer matrix; the fourth step of obtaining a total effective independence value of each of the sensors by summing up the obtained effective independence values of the respective paths; the fifth step of removing one having a smallest one of the obtained total effective independence values from among the sensors; and the sixth step of repeating the above second to fifth steps until the number of the remaining sensors becomes equal to a target value.
Preferably, the above target value may be set to be larger than a minimum number of the sensors, which is found by calculating variations in a conditional number of the transfer matrix while the sensors are reduced in number and detecting a point where the conditional number increases abruptly or a minimum point from a curved line obtained as a result of the calculation.
In a main feature of the present invention, there is provided a method for optimally determining sensor positions for an acoustic pyrometry, which is capable of obtaining a minimum number of sensors satisfying a given error limit and determining positions of the obtained number of sensors suitable for error minimization. Effective independence values are obtained from a transfer matrix representative of geometrical relations among sensor positions and propagation characteristics thereof and optimal sensor positions are determined on the basis of the obtained effective independence values. A desired minimum number of sensors is determined by observing variations in a condition number reflecting an error of the transfer matrix.